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Prime number theorem
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1120 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is \pi(N)~\frac{N}{\log(N)} , where \pi(N) is the prime-counting function and \log(N) is the natural logarithm of N''. This means that for large enough ''N, the probability that a random integer not greater than N'' is prime is very close to \frac{1}{\log(N)} . Consequently, a random integer with at most 2''n digits (for large enough n'') is about half as likely to be prime as a random integer with at most ''n digits. For example, among the positive integers of at most 1000 digits, about one in 2600 is prime ( \ log(10^{1000}) ≈ 2599.E035E8169131), whereas among positive integers of at most 2000 digits, about one in 5000 is prime ( \ log(10^{2000}) ≈ 4E77.X06EE4316262). In other words, the average gap between consecutive prime numbers among the first integers is roughly . Statement Let \pi(x) be the prime-counting function that gives the number of primes less than or equal to ''x, for any real number x''. For example, \pi(10)=5 because there are five prime numbers (2, 3, 5, 7 and E) less than or equal to 10. The prime number theorem then states that \frac{x}{\log(x)} is a good approximation to \pi(x) , in the sense that the limit of the ''quotient of the two functions \pi(x) and \frac{x}{\log(x)} as x'' increases without bound is 1: : \lim_{x\to\infty}\frac{\;\pi(x)\;}{\;\left[ \frac{x}{\log(x)}\right]\;} = 1, known as '''the asymptotic law of distribution of prime numbers'. Using asymptotic notation this result can be restated as : \pi(x)\sim \frac{x}{\log x}. This notation (and the theorem) does not say anything about the limit of the difference of the two functions as x'' increases without bound. Instead, the theorem states that \frac{x}{\log(x)} approximates \pi(x) in the sense that the relative error of this approximation approaches 0 as increases without bound. The prime number theorem is equivalent to the statement that the ''n''th prime number ''pn'' satisfies : p_n \sim n\log(n), the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as ''n increases without bound. For example, the 1010th prime number is 28,314,567,73E,4X1, and 10^{10}log(10^{10}) rounds to 25,99E,035,E81,691, a relative error of about E.2373%. The prime number theorem is also equivalent to : \lim_{x\to\infty} \frac{\vartheta (x)}x = \lim_{x\to\infty} \frac{\psi(x)}x=1, where and are the first and the second Chebyshev functions respectively. Table of \pi(x) , x/\log(x) , and li(x) \pi(N) ~ \frac{N}{log(N)} , where \pi(N) is the prime-counting function and log(N) is the natural logarithm of The last column, \frac{N}{\pi(N)} , is the average prime gap below N . li(x)=\int_0^x\frac{1}{\ln t}dt It has been conjectured that \pi(x) for all x, but this is not true (the first counterexample is around 10200). Category:Pages